I have a question about Comparison Geometry:
I have a Riemannian Manifold $(X,g)$, complete and simply connected, with sectional curvature upper bounded by a positive constant $k>0$, so I can compare $(X,g)$ with the sphere of ray $\frac{1}{\sqrt{k}}$ and constant curvature $k$, say $S_k$.
I want to show that $r_{cut}(x)\geqslant \frac{\pi}{\sqrt{k}}$ for every $x\in X$ in order to prove that diam$X$ $\geqslant$ diam$S_k$ (it would be enough to show that it's true for only one point and only one direction) and I'm trying to do it for absurd: if I suppose that for a point $x\in X$ and for a direction $u$ (i.e. $u\in T_x X$ and $|u|=1$) is $r_{cut}(x,u)<\frac{\pi}{\sqrt{k}}$ I know by some theorems that, if $y$ is the cut-point of $x$ along $u$, there are two different geodesics $\gamma_1$ and $\gamma_2$ joining $x$ and $y$, and by the simply connection hypothesis I can say that the two geodesics are homotopic relative to $\{0,1\}$. And then .... ? what can I say more?
Maybe there is another way for prove that diam$X$ $\geqslant$ diam$S_k$ ...